Question

Match each differential equation to a function which is a solution.

Answer 1

Answer: 1 - C

2 - E

3 - no answer

4 - B

Step-by-step explanation:

A. $y = 3x+x^2$

$y' = 3 + 2x\\y'' = 2$

• Replace in 1:

$y'' + y = 0$

$2 + 3x + x^2 \neq 0$

So, A is not an answer for 1

• Replace in 2:

$y' = 3y$

$3+2x = 3(3x + x^2)$

So, A is not an answer for 2

• Replace in 3

$2x^2y'' + 3xy'= y$

$2x^2(2) + 3x(3 + 2x) = 4x^2 + 9x + 6x^2 \neq 3x+x^2$

So, A is not an answer for 3

• Replace in 4

$y'' + 6y' + 8y = 0$

$2 + 6(3+2x)+8(3x + x^2) = 2+18+12x+24x+8x^2 \neq 0$

So, A is not an answer for 4

B. $y = e^{-4x}$

$y' = -4e^{-4x}$

$y'' = 16e^{-4x}$

• Replace in 1

$y'' + y = 0$

$16e^{-4x} -4e^{-4x} = 12e^{-4x} \neq 0$

So, B is not an answer for 1

• Replace in 2

$y' = 3y$

$-4e^{-4x} \neq 3e^{-4x}$

So, B is not an answer for 2

• Replace in 3

$2x^2y'' + 3xy' = y$

$2x^2(16e^{-4x}) +3x(-4e^{-4x}) \neq e^{-4x}$

So, B is not an answer for 3

• Replace in 4

$y'' + 6y' +8y = 0$

$16e^{-4x} + 6(-4e^{-4x}) + 8e^{-4x} = e^{-4x}(16-24+8) = 0$

So, B is an answer for 4

C. $y = sin(x)$

$y' = cos(x)$

$y'' = -sin(x)$

• Replace in 1

$y'' + y = 0$

$-sin(x) + sin(x) = 0$

So, C is an answer for 1

We jump to

D. $y = x^{12}$

$y' = 12x^{11}$

$y'' = 132 x^{10}$

• Replace in 2

$y' = 3y$

$12x^{11} \neq 3x^{12}$

So, D is not an answer for 2

• Replace in 3

$2x^2y'' + 3xy' = y$

$2x^2(132x^{10}) + 3x(12x^{11}) = 300x^{12} \neq x^{12}$

So, D is not an answer for 3

E. $y = 6e^{3x}$

$y' = 18e^{3x}$

$y'' = 54e^{3x}$

• Replace in 2

$y' = 3y$

$18e^{3x} = 3(6e^{3x}) = 18e^{3x}$

So, E is an answer for 2